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Cone Frustum Calculator (Truncated Cone)

Volume equals pi h divided by three times the quantity capital R squared plus capital R times r plus r squared

Solution

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Calculate Cone Frustum Volume from R, r, and h

Use this form when both base radii and the perpendicular height of a truncated cone are known. The volume reduces to a cylinder (R = r) at one extreme and to a full cone (r = 0) at the other.

V = (π h / 3)(R² + R r + r²)

Calculate Total Surface Area of a Cone Frustum

Use this form when you need the entire exterior — both circular ends plus the slanted side. Common for closed buckets, lampshade end caps, and material takeoffs on hoppers with a top flange.

S = π (R + r) ℓ + π R² + π r²

Calculate Lateral Surface Area of a Cone Frustum

Use this form when only the slanted side matters — the wrap on a lampshade, the sheet metal needed to roll a hopper, or the paint coverage on the curved face of a bucket.

L = π (R + r) ℓ

Calculate Cone Frustum Height from V, R, and r

Use this rearrangement when a target volume and both radii are fixed and you need the perpendicular height. Common when sizing a bucket or hopper to a specified capacity for given rim dimensions.

h = 3V / (π (R² + R r + r²))

How It Works

This cone frustum calculator (truncated cone) uses V = (π h / 3)(R² + R r + r²) plus the surface-area pair S = π (R + r) ℓ + π R² + π r² and L = π (R + r) ℓ, where ℓ = √(h² + (R − r)²) is the slant height. Pick the unknown with the solve-for toggle, enter the remaining three values in any supported length or volume unit, and the calculator converts to SI internally before computing every related quantity — volume, both surface areas, slant height, both radii, and the perpendicular height — so a single page covers buckets, lampshades, conical hoppers, drinking cups, and frusta in geometry homework.

Example Problem

A small bucket has a bottom radius R = 3 m, a top radius r = 1 m, and a height h = 4 m. What is its volume, slant height, lateral surface area, and total surface area?

  1. Identify the dimensions: R = 3 m, r = 1 m, h = 4 m.
  2. Compute the slant height ℓ = √(h² + (R − r)²) = √(16 + 4) = √20 = 2√5 ≈ 4.4721 m.
  3. Volume: V = (π h / 3)(R² + R r + r²) = (4π/3) · (9 + 3 + 1) = 52π/3 ≈ 54.454 m³.
  4. Lateral surface area: L = π (R + r) ℓ = π · 4 · 2√5 = 8π√5 ≈ 56.1985 m².
  5. Total surface area: S = L + π R² + π r² = 8π√5 + 9π + π = 8π√5 + 10π ≈ 87.6144 m².
  6. Round trip: from V ≈ 54.454, R = 3, r = 1, the inverse h = 3V / (π (R² + R r + r²)) recovers h = 4 m, confirming the volume.

When r = 0 the frustum collapses into a full cone — and V = 12π, ℓ = 5, L = 15π, S = 24π, matching the cone formulas exactly. When R = r the frustum is a cylinder with ℓ = h.

When to Use Each Variable

  • Solve for Volume — when R, r, and h are known and you need the capacity of a bucket, hopper, drinking cup, or lampshade-shaped container.
  • Solve for Total Surface Area — when you need the entire exterior including both flat ends — common for closed buckets and capped hoppers.
  • Solve for Lateral Surface Area — when only the slanted side matters — lampshade wraps, sheet-metal patterns, paint or label coverage on a bucket.
  • Solve for Height — when both radii are fixed by available stock and you need the height that gives a target volume.

Key Concepts

A cone frustum — sometimes just called a truncated cone — is what you get when you slice a right circular cone with a plane parallel to its base and remove the small cone on top. Three numbers determine the geometry: the bottom radius R, the top radius r, and the perpendicular height h. The slant height ℓ = √(h² + (R − r)²) plays the same role here that it does for a cone — it scales the lateral surface and is the actual length of the slanted edge in cross section. The volume formula V = (π h / 3)(R² + R r + r²) interpolates smoothly between two familiar cases: when r → 0 it becomes (1/3) π R² h (a full cone); when r → R it becomes π R² h (a cylinder). The mixed term R r is what makes the average of the two end-area approximations work exactly.

Applications

  • Buckets, pails, and flowerpots: compute capacity from rim and base diameters and depth
  • Lampshades and pendant lights: estimate wrap material and silhouette dimensions
  • Hoppers and silos: size conical transition sections between cylindrical bins and discharge openings
  • Drinking cups and conical glasses: convert volume markings between fluid-ounce and liter scales
  • Sheet-metal fabrication: lay out the flat pattern for a frustum (an annular sector when unrolled)
  • Geometry instruction: a textbook shape that connects the cone and cylinder formulas

Common Mistakes

  • Using cone height h in place of slant height ℓ in the lateral-area formula — L = π(R+r)ℓ uses the slant, not the perpendicular height
  • Forgetting the R r mixed term in the volume formula — V = (π h / 3)(R² + r²) is wrong; the correct sum is R² + R r + r²
  • Computing slant height as √(R² + h²) (the cone formula) instead of √(h² + (R − r)²) — the difference R − r is what matters, not R alone
  • Confusing R and r — by convention R is the larger (bottom) radius and r is the smaller (top), though the volume and slant formulas are symmetric in the two radii
  • Including only one end disk in the total surface area — both the top π r² and bottom π R² disks count when the frustum is closed at both ends

Frequently Asked Questions

How do you calculate the volume of a cone frustum (truncated cone)?

Use V = (π h / 3)(R² + R r + r²), where R is the bottom radius, r is the top radius, and h is the perpendicular height. For example, R = 3 m, r = 1 m, h = 4 m gives V = (4π/3) · 13 = 52π/3 ≈ 54.5 m³.

What is the formula for the slant height of a frustum?

Slant height ℓ = √(h² + (R − r)²). It is the actual length of the slanted edge in cross section, not the perpendicular height. For R = 3, r = 1, h = 4 this gives ℓ = √20 = 2√5 ≈ 4.47.

What is the difference between total and lateral surface area for a frustum?

Lateral surface area L = π (R + r) ℓ counts only the slanted side. Total surface area S = L + π R² + π r² adds both circular ends. Use lateral area for a lampshade wrap or open bucket; use total surface area for a closed container.

How do I size a conical hopper using this calculator?

Set the bottom radius R to your discharge-spout radius, the top radius r to your bin radius, and the height h to the vertical transition. The calculator returns the hopper volume and the slant height needed for the sheet-metal pattern.

How do I find the height of a frustum given its volume?

Rearrange the volume formula to h = 3V / (π (R² + R r + r²)). For V = 54.5 m³, R = 3 m, r = 1 m: h = 3 · 54.5 / (π · 13) ≈ 4 m.

Does a cone frustum become a full cone when the top radius is zero?

Yes — set r = 0 and the formulas collapse to the cone formulas: V = (1/3) π R² h, ℓ = √(R² + h²), L = π R ℓ, S = L + π R². The frustum calculator and the cone calculator agree exactly in that limit.

How do I calculate the capacity of a bucket or lampshade?

Measure the inside diameter at the bottom and top, divide each by 2 to get R and r, measure the inside depth h, and plug into V = (π h / 3)(R² + R r + r²). The same formula works for a pail, planter, drinking glass, or lampshade silhouette.

How is a cone frustum unrolled into a flat sheet?

The lateral surface unrolls into an annular sector — a ring-shaped slice of a circle with inner arc length 2π r and outer arc length 2π R, separated radially by the slant height ℓ. Sheet-metal shops use this pattern to roll buckets, hoppers, and lampshade frames.

Reference: Weisstein, Eric W. "Conical Frustum." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/ConicalFrustum.html

Worked Examples

Bucket Capacity

How do you calculate the volume of a tapered bucket?

A small bucket has a bottom radius R = 3 m, top radius r = 1 m, and a height h = 4 m (the canonical Wolfram case). Use V = (π h / 3)(R² + R r + r²) for the capacity.

  • Knowns: R = 3 m, r = 1 m, h = 4 m
  • Formula: V = (π h / 3) · (R² + R r + r²)
  • V = (4π/3) · (9 + 3 + 1) = (4π/3) · 13
  • V = 52π/3 ≈ 54.454 m³
  • Convert to liters: 54.454 m³ × 1000 ≈ 54,454 L

Volume ≈ 54.5 m³ ≈ 54,454 liters

Real buckets have wall thickness — for working volume measure interior diameters and depth, not exterior.

Lampshade Wrap

How much fabric is needed to wrap a tapered lampshade?

A pendant lampshade has a bottom (skirt) radius R = 25 cm, top (collar) radius r = 12 cm, and height h = 30 cm. The fabric needed is the lateral surface area L = π(R + r)ℓ.

  • Knowns: R = 25 cm, r = 12 cm, h = 30 cm
  • Slant height ℓ = √(h² + (R − r)²) = √(900 + 169) = √1069 ≈ 32.70 cm
  • Formula: L = π (R + r) ℓ
  • L = π · 37 · 32.70 ≈ 3,801 cm² ≈ 0.380 m² (≈ 4.09 ft²)

Lateral surface area ≈ 3,801 cm² (~0.38 m²)

When rolled flat, the shade unwraps into a circular sector with arc lengths 2πR and 2πr separated by ℓ — that's the pattern the shop cuts. Add a small seam-allowance on the cut edge.

Grain Hopper

What height does a 500-liter grain hopper need with fixed rim radii?

A conical-transition hopper has bin radius R = 0.6 m at the top of the transition and discharge radius r = 0.1 m at the bottom. The target capacity is 500 L (0.5 m³). Solve h = 3V / (π(R² + R r + r²)).

  • Knowns: V = 0.5 m³, R = 0.6 m, r = 0.1 m
  • Compute R² + R r + r² = 0.36 + 0.06 + 0.01 = 0.43 m²
  • Formula: h = 3V / (π · (R² + R r + r²))
  • h = 3 · 0.5 / (π · 0.43) ≈ 1.5 / 1.3509 ≈ 1.110 m

Hopper height ≈ 1.11 m (≈ 43.7 in)

Note that R and r in the volume formula are interchangeable — flipping which end is wider doesn't change the volume, only the orientation. Real hoppers also factor in the discharge angle (slope of the slant) for material-flow purposes.

Cone Frustum Formulas

All cone-frustum properties follow from three dimensions: the bottom radius R, the top radius r, and the perpendicular height h. From those, slant height, volume, and both surface areas all derive:

ℓ = √(h² + (R − r)²)
V = (π h / 3) · (R² + R r + r²)
L = π (R + r) · √(h² + (R − r)²)
S = π (R + r) · √(h² + (R − r)²) + π R² + π r²
h = 3V / (π (R² + R r + r²))
Cone frustum with labeled bottom radius R, top radius r, and height hRrh

Where:

  • V — enclosed volume (m³, L, gal, ft³)
  • S — total surface area: both circular ends plus the slanted side (m², ft², in²)
  • L — lateral surface area: curved side only, unrolls into an annular sector
  • R — bottom radius, conventionally the larger of the two (m, cm, in, ft, yd)
  • r — top radius, conventionally the smaller; setting r = 0 reduces the frustum to a full cone
  • h — perpendicular height between the two parallel circular ends
  • ℓ — slant height, the apex-to-rim distance along the lateral surface: ℓ = √(h² + (R − r)²)

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